Euclidean Wormholes and Connected Boundary Amplitudes

A Euclidean wormhole is a connected Euclidean saddle with more than one asymptotic boundary. If its boundary conditions, topology policy, integration cycle, fluctuation spectrum, and moduli measure admit it, it contributes to a connected multi-boundary amplitude. That fact alone establishes neither a Lorentzian traversable geometry nor an ensemble interpretation.

Required background. Euclidean Gravitational Path Integrals, Topology Sums, and Boundary Conditions defines the amplitude. Saddles, Negative Modes, and Steepest-Descent Cycles determines whether a saddle actually lies on the contour.

Helpful background. Non-Unique JT Matrix-Integral Completion: Fixed-Theory, Disorder-Average, and Ensemble Distinctions supplies a controlled ensemble comparison. Renormalized Saddle Contributions and Validity Tests supplies general reliability checks.

Connected versus disconnected boundary amplitudes

For two Euclidean boundaries $B_1$ and $B_2$, define

$$Z[B_1,B_2]= \sum_{M:\,\partial M=B_1\sqcup B_2} \int_{\mathcal C_M}\frac{\mathcal Dg\,\mathcal D\Phi}{\mathrm{Diff}_0} e^{-I_E[g,\Phi]} .$$

If both disconnected $M_1\sqcup M_2$ and connected $M_{12}$ topologies are allowed, then at saddle order

$$Z[B_1,B_2]\simeq Z[B_1]Z[B_2] +e^{-I_{\mathrm{ren}}[g_{12}]} \frac{\det{}'\Delta_{\mathrm{gh},12}} {\sqrt{\det{}'\Delta_{\mathrm{phys},12}}} \int_{\mathcal M_{12}}d\mu+\cdots .$$

The last factors schematically denote one-loop determinants and a moduli integral; their normalization is part of the claim. The connected part is

$$Z_{\mathrm{conn}}[B_1,B_2] =Z[B_1,B_2]-Z[B_1]Z[B_2].$$

It is a property of the gravitational prescription just defined. Calling it a covariance already assumes an averaging measure.

Application: the JT double trumpet

In JT gravity, the genus-zero connected surface with two asymptotic boundaries is the double trumpet. Cutting along its minimal geodesic of length $b$ gives two trumpets and a Weil–Petersson gluing measure:

$$Z_{0,2}(\beta_1,\beta_2) =\int_0^\infty b\,db\, Z_{\mathrm{tr}}(\beta_1,b)Z_{\mathrm{tr}}(\beta_2,b).$$

With the conventional JT trumpet normalization

$$Z_{\mathrm{tr}}(\beta,b) =\frac{1}{\sqrt{2\pi\beta}} \exp\!\left(-\frac{b^2}{2\beta}\right),$$

the integral is finite:

$$Z_{0,2}(\beta_1,\beta_2) =\frac{\sqrt{\beta_1\beta_2}}{2\pi(\beta_1+\beta_2)}.$$

The Euler characteristic of the cylinder is zero, so this term is not multiplied by the disk factor $e^{S_0}$. In the Saad–Shenker–Stanford matrix-integral completion it equals a connected ensemble correlator of partition functions Saad, Shenker, and Stanford 2019. That interpretation follows from the independently supplied matrix ensemble, not from connectedness alone.

Contour and normalization check

Now make one adversarial change while holding $\beta_1,\beta_2$ fixed.

• Exclude connected topology: $Z_{0,2}$ vanishes by definition.
• Choose a contour whose intersection number with the connected saddle is zero: the formal solution remains but contributes nothing.
• Find an unpaired physical negative mode: the naive real positive expression is not the amplitude until its descent cycle is fixed.
• Change the twist-period or trumpet normalization: the coefficient changes even though the topology and exponential scaling do not.

This check prevents a geometry-only argument. Higher-dimensional Euclidean wormholes face additional stability and boundary-source issues; known smooth saddles can be unstable or require special matter and boundary conditions Maldacena and Maoz 2004.

What the result licenses

A nonzero connected term licenses the statement that the declared gravitational path integral is nonfactorizing at that approximation. It does not by itself identify whether the dual object is an ensemble average, a conditioned baby-universe sector, an incomplete low-energy description, or a fixed theory with omitted contributions that restore factorization. Those alternatives are tested on Factorization, Ensembles, and the Gravitational Path Integral.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Maldacena, J., and L. Maoz. “Wormholes in AdS.” Journal of High Energy Physics 2004, 2 (2004): 053. DOI.
• Saad, P., S. H. Shenker, and D. Stanford. “JT Gravity as a Matrix Integral.” arXiv:1903.11115.